Summary: In 2017, Bo’az Klartag obtained a new result in differential geometry on the existence of affine hemisphere of elliptic type. In his approach, a surface is associated with every convex function \(\varphi : \mathbb{R}^n\to(0,+\infty)\) and the condition for the surface to be an affine hemisphere involves the \(2\)-moment measure of \(\varphi\) (a particular case of \(q\)-moment measures, i.e measures of the form \((\nabla\varphi)_\#\varphi^{-(n+q)}\) for \(q > 0\)). In Klartag’s paper, \(q\)-moment measures are studied through a variational method requiring to minimize a functional among convex functions, which is achieved using the Borell-BrascampLieb inequality. In this paper, we attack the same problem through an optimal transport approach, since the convex function \(\varphi\) is a Kantorovich potential (as already done for moment measures in a previous paper). The variational problem in this new approach becomes the minimization of a local functional and a transport cost among probability measures \(\varrho\) and the optimizer \(\varrho_\mathrm{opt}\) turns out to be of the form \(\varrho_\mathrm{opt}=\varphi^{-(n+q)}\)